Atmospheric Characterization of Hot Jupiters Using Hierarchical Models of Spitzer Observations

MNRAS 000,1–10 (2021) Preprint 2 March 2021 Compiled using MNRAS LATEX style file v3.0 Atmospheric characterization of hot Jupiters using hierarchical models of Spitzer observations Dylan Keating,1¢ Nicolas B. Cowan,1,2 1Department of Physics, McGill University, Montréal, QC H3A 2T8, Canada 2Department of Earth & Planetary Sciences, McGill University, Montréal, QC H3A 2T8, Canada Submitted 2021 February 25 ABSTRACT The field of exoplanet atmospheric characterization is trending towards comparative studies involving many planets, and using hierarchical modelling is a natural next step. Here we demonstrate two use cases. We first use hierarchical modelling to quantify variability in repeated observations, by reanalyzing a suite of ten Spitzer secondary eclipse observations of the hot Jupiter XO-3b. We compare three models: one where we fit ten separate eclipse depths, one where we use a single eclipse depth for all ten observations, and a hierarchical model. By comparing the Widely Applicable Information Criterion (WAIC) of each model, we show that the hierarchical model is preferred over the others. The hierarchical model yields less scatter across the suite of eclipse depths, and higher precision on the individual eclipse depths, than does fitting the observations separately. We do not detect appreciable variability in the secondary eclipses of XO-3b, in line with other analyses. Finally, we fit the suite of published dayside brightness measurements from Garhart et al.(2020) using a hierarchical model. The hierarchical model gives tighter constraints on the individual brightness temperatures and is a better predictive model, according to the WAIC. Notably, we do not detect the increasing trend in brightness temperature ratios versus stellar irradiation reported by Garhart et al.(2020) and Baxter et al.(2020). Although we tested hierarchical modelling on Spitzer eclipse data of hot Jupiters, it is applicable to observations of smaller planets like hot neptunes and super earths, as well as for photometric and spectroscopic transit or phase curve observations. Key words: planets and satellites: individual (XO-3b) – techniques: photometric 1 INTRODUCTION from multiple planets simultaneously using hierarchical models to robustly infer trends. Although the Spitzer Space Telescope wasn’t designed for exoplanet science, it was a workhorse for the field. In particular, observations of exoplanet transits, secondary eclipses, and phase curves with Spitzer’s Infrared Array Camera have been used to characterize the atmospheres of over a hundred transiting planets. 1.1 Spitzer Systematics Substantial progress has been made towards a statistical under- Exoplanet observations taken with Spitzer’s Infrared Array Camera standing of exoplanetary atmospheres (Cowan & Agol 2011; Sing (IRAC; Fazio et al.(2004)) are dominated by systematics noise. et al. 2016; Schwartz & Cowan 2015; Schwartz et al. 2017; Parmen- The systematics are driven by intrapixel sensitivity variations on the tier & Crossfield 2018; Zhang et al. 2018; Keating et al. 2019, 2020; detector and by now are well characterized (Ingalls et al. 2016). Baxter et al. 2020; Bell et al. 2020). Many of the planets in these arXiv:2103.00010v1 [astro-ph.EP] 26 Feb 2021 The detector systematics are typically fitted simultaneously with the studies had been analyzed using disparate reduction and analysis astrophysical signal of interest. Each transit, secondary eclipse, and pipelines, but recently the field has been moving towards analyzing phase curve yields information about the IRAC detector sensitivity, suites of observations from multiple planets using a single pipeline. but typically this information is not shared between observations. Garhart et al.(2020) independently reduced and analyzed 78 eclipse Since the Spitzer systematics are a function of the centroid loca- depths from 36 planets and found that hotter planets had higher tion on the pixel, efforts have been made to determine the detector brightness temperatures at 4.5 `m than at 3.6 `m. Bell et al.(2020) sensitivity independently using observations of quiet stars (Ingalls reanalyzed every available Spitzer 4.5 `m hot Jupiter phase curve et al. 2012; Krick et al. 2020; May & Stevenson 2020). The flux of using an open-source reduction and analysis pipeline, confirming a calibration star should be constant as a function of time, so any several previously reported trends. deviation must be due to the centroid moving across the detector as In this work we outline a complementary way to further the statis- the telescope pointing drifts. A crucial assumption for this approach tical understanding of exoplanet atmospheres: fitting measurements is that the Spitzer systematics do not vary with time, and that they are not dependent on the brightness of the star. Ingalls et al.(2012) and May & Stevenson(2020) approached the problem by explicitly ¢ E-mail: [email protected] calculating the detector sensitivity, while Krick et al.(2020) used a © 2021 The Authors 2 D. Keating and N. Cowan machine learning technique called random forests to look for patterns underfitting all of the observations. If we observe one secondary in the systematics. eclipse, we would expect that the next one we observe would have a Other approaches do not assume anything explicit about the detec- similar— but not identical— depth, due to measurement uncertainty, tor sensitivity. Independent component analysis (Waldmann 2012; if not astrophysical variability. The second eclipse we observe would Morello et al. 2014, 2016) separates the signal into additive subcom- also change our beliefs about the first one. Each measurement of the ponents using blind source separation, with the idea being that one planet’s eclipse depth can be thought of as a draw from a distribution, of these signals is the astrophysical signal. In another approach, Mor- with some variance. With enough measurements, the shape of this van et al.(2020) used the baseline signal before and after a transit to distribution can be inferred. A hierarchical model naturally takes all learn and predict the in transit detector systematics using a machine of this into account by fitting for the parameters that describe the learning technique known as Long short-term memory networks. higher level distribution simultaneously with the astrophysical signal In this work, we opted to model and fit the detector systematics to of each observation. allow for any correlations between the systematics and astrophysical Bayesian analysis requires us to specify priors on the parameters signals. we are trying to infer. We can write down our prior on the 8th eclipse depth as 1.2 Hierarchical Models 퐸퐷8 ∼ N ¹`, fº, (1) Hierarchical models (Gelman et al. 2014) are routinely used in other where we have used shorthand to say that the eclipse depth is drawn fields because they offer a natural way to infer higher level trends that from a normal distribution centered on ` with a standard deviation exist in a dataset, and can increase measurement precision. They are of f; ` and f are hyperparameters. In a non-hierarchical model, we gaining traction in exoplanet studies: for example, to study the mass- would specify values ` and f to represent our prior expectations of radius (Teske et al. 2020) and mass-radius-period (Neil & Rogers what 퐸퐷8 could be. After fitting, we would get a separate posterior 2020) relations, and radius inflation of hot Jupiters (Sarkis et al. distribution for each eclipse depth. 2021; Thorngren et al. 2021). Hierarchical models have not yet been In a hierarchical model, we instead make ` and f parameters and applied to atmospheric characterization of exoplanets. fit them simultaneously with the ten eclipse depths. We represent our There is one major difference between a typical Bayesian model beliefs about hyperparameters ` and f with hyperpriors. This allows and a hierarchical one. In a hierarchical model one assumes that each eclipse observation to inform the others, by pulling the eclipse some of the model parameters of interest are drawn from a higher depths closer to the mode of the distribution of `. This is known as level, unobserved distribution. The higher level parameters describ- Bayesian shrinkage. After fitting, we get a posterior distribution for ing this distribution, otherwise known as hyperparameters, are fitted each eclipse depth, as well as for ` and f. simultaneously with the parameters of interest. As we explain below, In the limit that f goes to infinity, the hierarchical model is equiv- this naturally represents how our intuition pools information across alent to the model with completely separate eclipse depths. Likewise observations. It also helps to tame models by compromising between when f goes to zero, it is equivalent to the single eclipse depth model. overfitting and underfitting. A hierarchical model empirically fits for the amount of pooling based Hierarchical models can and should be used whenever a higher on what is most consistent with the observations. level structure can be assumed in a data set, such as when one quantity is measured multiple times. A natural example is the archetypical suite of of ten XO-3b Spitzer IRAC Channel 2 (4.5 `m) eclipses 2.1 Priors (Wong et al. 2014). Below we explain the model and present results. Priors are necessary in a model to encode prior knowledge, as well Afterwards, we show how we extend the model to fit multiple eclipses as to properly sample a model. In all cases, we use weakly infor- from different planets simultaneously and present results from fitting mative priors rather than flat, “uninformative” priors. Half-normal the eclipse data from Garhart et al.(2020) with a hierarchical model. priors or wide normal priors are unlikely to introduce much bias into the parameter estimates and can make sampling more efficient. Flat priors are discouraged in practice because we usually have at least 2 HIERARCHICAL MODEL OF XO-3B ECLIPSES some vague knowledge of the range of values a parameter can take (Gelman et al. 2017). In the Spitzer data challenge, several groups analyzed ten secondary For instance, consider a flat prior on the eclipse depth.

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